Spectral analysis and exponential stability of a generalized fractional Moore-Gibson-Thompson equation

Flank Bezerra; Lucas Santos; Maria Silva; Carlos Takaessu Jr

doi:10.3934/dcdsb.2024098

Article Contents

2025, Volume 30, Issue 2: 496-508. Doi: 10.3934/dcdsb.2024098

This issue Previous Article Uniqueness of global conservative weak solutions for two-component higher-order shallow water system Next Article Stability of a thermo-elastic system of type Ⅱ with fractional boundary damping

Spectral analysis and exponential stability of a generalized fractional Moore-Gibson-Thompson equation

1.
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brazil
2.
Instituto Federal da Paraíba, 58051-900 João Pessoa PB, Brazil
3.
Universidade de São Paulo, ICMC, Av. Trabalhador São-Carlense, 13566590, São Carlos, SP, Brazil

^*Corresponding author: Flank Bezerra
^*Corresponding author: Flank Bezerra

Received: January 2024

Revised: June 2024

Early access: July 22, 2024

Published online: July 22, 2024

The first author is supported by [CNPq/Finance Code # 303039/2021-3]. The last author is supported by [CAPES-PROEX-11169228/D and by FAPESP # 2020/14353-6, Brazil].

Abstract / Introduction Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by

Abstract

In this paper, we consider a class of abstract third-order linear evolution equations in time which can be seen as a generalized fractional Moore-Gibson-Thompson (MGT) equation. Given the classic MGT model, we prove the well-posedness and exponential stability of our model in a suitable phase space.

Keywords:

Mathematics Subject Classification: Primary: 35R11, 47A08.

Citation:

Full Text(HTML)

Table 1. $ \alpha b<c^2 $ and $ \alpha b\geq c^2 $.

	Analytic Semigroup	Strongly Continuous Semigroup
$ \alpha b <c^2 $	x	x
$ \alpha b\geq c^2 $	x	v

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. E. Abouelregal, H. M. Sedighi and V. A. Eremeyev, Thermomagnetic behavior of a semiconductor material heated by pulsed excitation based on the fourth-order MGT photothermal model, Continuum Mech. Thermodyn., 35 (2023), 81-102. doi: 10.1007/s00161-022-01170-z.
[2]	H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.
[3]	F. D. M. Bezerra, A. N. Carvalho, J. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: Fast-growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl., 450 (2017), 377-405. doi: 10.1016/j.jmaa.2017.01.024.
[4]	F. D. M. Bezerra and L. A. Santos, Fractional powers approach of operators for abstract evolution equations of third-order in time, J. Differential Equations, 269 (2020), 5661-5679. doi: 10.1016/j.jde.2020.04.020.
[5]	A. H. Caixeta, I. Lasiecka and V. N. D. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147. doi: 10.1016/j.jde.2016.03.006.
[6]	R. Datko, Extending a theorem of A. M. Liapunov to Hilbert spaces, J. Math. Anal. Appl., 32 (1970), 610-616. doi: 10.1016/0022-247X(70)90283-0.
[7]	Ł. Dawidowski, Scales of Banach Spaces, Theory of Interpolation and their Applications, Wydawnictwo Uniwersytetu Slaskiego, PhD Thesis, Katowice, 2012.
[8]	F. Dell'Oro, L. Liverani and V. Pata, On the regularized Moore-Gibson-Thompson equation, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 2326-2338. doi: 10.3934/dcdss.2023025.
[9]	F. Dell'Oro and V. Pata, On a fourth-order equation of Moore-Gibson-Thompson type, Milan J. Math., 85 (2017), 215-234. doi: 10.1007/s00032-017-0270-0.
[10]	L. Gui-Zhong, Evolution Equation and Scales of Banach Spaces, Technische Universiteit Eindhoven, Phd Thesis, 1989.
[11]	M. Hasse, The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl., 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.
[12]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.
[13]	B. Kaltenbacher, I. Lasiecka and R. Marchand, Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet, 40 (2011), 971-988.
[14]	B. Kaltenbacher and V. Nikolić, Time-fractional Moore-Gibson-Thompson equations, Math. Models Methods Appl. Sci., 32 (2022), 965-1013. doi: 10.1142/S0218202522500221.
[15]	S. G. Kreǐn, Linear Differential Equations in a Banach Space, Transl. Mathem. Monogr., 29, Amer. Math. Soc., 1971.
[16]	R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929. doi: 10.1002/mma.1576.
[17]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[18]	M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Applied Mathematics & Optimization, 80 (2019), 447-478. doi: 10.1007/s00245-017-9471-8.
[19]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, 1978.
[20]	E. B. Vinberg, A Course in Algebra, Graduate Studies in Mathematics, Vol. 56, American Mathematical Soc., Providence, Rhode Island, 2003. doi: 10.1090/gsm/056.
[21]	K. Yosida, Functional Analysis, Springer, Berlin, 1978. doi: 10.1007/978-3-642-96439-8.